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Resonance and nonlinearity: A survey

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Ukrainian Mathematical Journal Aims and scope

Abstract

This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and piecewise-linear oscillators, as well as of some classes of planar Hamiltonian systems.

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References

  1. J. M. Alonso and R. Ortega, “Unbounded solutions of semilinear equations at resonance,” Nonlinearity, 9, 1099–1111 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  2. J. M. Alonso and R. Ortega, “Roots of unity and unbounded motions of an asymmetric oscillator,” J. Different. Equat., 143, 201–220 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  3. N. N. Bogolyubov and Yu. A. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1961).

    Google Scholar 

  4. D. Bonheure and C. Fabry, “Unbounded solutions of forced isochronous oscillators at resonance,” Different. Integr. Equat., 15, 1139–1152 (2002).

    MATH  MathSciNet  Google Scholar 

  5. D. Bonheure and C. Fabry, Littlewood’s Problem for Isochronous Oscillations (to appear).

  6. D. Bonheure, C. Fabry, and D. Smets, “Periodic solutions of forced isochronous oscillators at resonance,” Discrete Contin. Dynam. Syst., 8, 907–930 (2002).

    MATH  MathSciNet  Google Scholar 

  7. A. Capietto and Bin Liu, Quasi-Periodic Solutions of a Forced Asymmetric Oscillator at Resonance (to appear).

  8. A. Capietto, J. Mawhin, and F. Zanolin, “Continuation theorems for periodic perturbations of autonomous systems,” Trans. Amer. Math. Soc., 329, 41–72 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Capietto and Zaihong Wang, “Periodic solutions of Liénard equations at resonance,” Different. Integr. Equat., 16, 605–624 (2003).

    MATH  MathSciNet  Google Scholar 

  10. A. Capietto and Zaihong Wang, “Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,” J. London Math. Soc., 68, 119–132 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Dambrosio, “A note on the existence of unbounded solutions to a perturbed asymmetric oscillator,” Nonlin. Analysis, 50, 333–346 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. E. N. Dancer, “Boundary-value problems for weakly nonlinear ordinary differential equations,” Bull. Austral. Math. Soc., 15, 321–328 (1976).

    MATH  MathSciNet  Google Scholar 

  13. E. N. Dancer, “Proofs of the results of ‘boundary-value problems for weakly nonlinear ordinary differential equations’” (unpublished).

  14. C. Fabry, “Landesman-Lazer conditions for periodic boundary value problems with asymmetric nonlinearities,” J. Different. Equat., 116, 405–418 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Fabry and A. Fonda, “Nonlinear resonance in asymmetric oscillators,” J. Different. Equat., 147, 58–78 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Fabry and A. Fonda, “Periodic solutions of perturbed isochronous Hamiltonian systems at resonance,” J. Different. Equat., 214, 299–325 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  17. C. Fabry and A. Fonda, “Unbounded motions of perturbed isochronous Hamiltonian systems at resonance,” Adv. Nonlin. Stud., 5, 351–373 (2005).

    MATH  MathSciNet  Google Scholar 

  18. C. Fabry and J. Mawhin, “Oscillations of a forced asymmetric oscillator at resonance,” Nonlinearity, 13, 493–505 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Fabry and J. Mawhin, “Properties of solutions of some forced nonlinear oscillators at resonance,” in: K. C. Chang and Y. M. Long (editors), Nonlinear Analysis, World Scientific, Singapore (2000), pp. 103–118.

    Google Scholar 

  20. A. Fonda, “Positively homogeneous Hamiltonian systems in the plane,” J. Different. Equat., 200, 162–184 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Fonda, “Topological degree and generalized asymmetric oscillators,” Top. Meth. Nonlin. Analysis, 28, 171–188 (2006).

    MATH  MathSciNet  Google Scholar 

  22. A. Fonda and J. Mawhin, “Planar differential systems at resonance,” Adv. Different. Equat. (to appear).

  23. S. Fučík, “Boundary value problems with jumping nonlinearities,” Č as. Péstov. Mat., 101, 69–87 (1976).

    MATH  Google Scholar 

  24. S. Fučík, Solvability of Nonlinear Equations and Boundary Value Problems, Reidel, Boston (1980).

    MATH  Google Scholar 

  25. R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin (1977).

    MATH  Google Scholar 

  26. M. Kunze, “Remarks on boundedness of semilinear oscillators,” in: Nonlinear Analysis and Its Applications to Differential Equations (Lisbon, 1998), Birkhauser, Boston (2001), pp. 311–319.

    Google Scholar 

  27. M. Kunze, T. Kupper, and Bin Liu, “Boundedness and unboundedness of solutions of reversible oscillators at resonance,” Nonlinearity, 14, 1105–1122 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  28. A. C. Lazer and D. E. Leach, “Bounded perturbations of forced harmonic oscillators at resonance,” Ann. Mat. Pura Appl., 82, 49–68 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  29. Xiong Li, “Boundedness of solutions for semilinear reversible systems,” Proc. Amer. Math. Soc., 132, 2057–2066 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  30. Xiong Li, “Invariant tori for semilinear reversible systems,” Nonlin. Analysis, 56, 133–146 (2004).

    Article  MATH  Google Scholar 

  31. Xiong Li and Qing Ma, “Boundedness of solutions for second order differential equations with asymmetric nonlinearity,” J. Math. Anal. Appl., 314, 233–253 (2006).

    MATH  MathSciNet  Google Scholar 

  32. J. E. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington (1969).

    Google Scholar 

  33. Bin Liu, “Boundedness of solutions for semilinear Duffing equations,” J. Different. Equat., 145, 119–144 (1998).

    Article  MATH  Google Scholar 

  34. Bin Liu, “Invariant tori in nonlinear oscillations,” Sci. China A, 42, 1047–1058 (1999).

    Google Scholar 

  35. Bin Liu, “Boundedness in asymmetric oscillations,” J. Math. Anal. Appl., 231, 355–373 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  36. Bin Liu, “Boundedness in nonlinear oscillations at resonance,” J. Different. Equat., 153, 142–174 (1999).

    Article  MATH  Google Scholar 

  37. Bin Liu, “Quasiperiodic solutions of semilinear Liénard equations,” Discrete Contin. Dynam. Syst., 12, 137–160 (2005).

    Article  MATH  Google Scholar 

  38. A. I. Lur’e, Some Nonlinear Problems in the Theory of Automatic Control [in Russian], Gostekhizdat, Moscow-Leningrad (1951).

    Google Scholar 

  39. J. L. Massera, “The existence of periodic solutions of systems of differential equations,” Duke Math. J., 17, 457–475 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  40. G. R. Morris, “A case of boundedness of Littlewood’s problem on oscillatory differential equations,” Bull. Austral. Math. Soc., 14, 71–93 (1976).

    MATH  MathSciNet  Google Scholar 

  41. J. Moser, “On invariant curves of area-preserving mappings of annulus,” Nachr. Akad. Wiss. Gottingen. II. Math.-Phys. Kl., 1–20 (1962).

  42. R. Ortega, “Asymmetric oscillators and twist mappings,” J. London Math. Soc., 53, 325–342 (1996).

    MATH  MathSciNet  Google Scholar 

  43. R. Ortega, “On Littlewood’s problem for the asymmetric oscillator,” Rend. Semin. Mat. Fis. Milano, 68, 153–164 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  44. R. Ortega, “Boundedness in a piecewise linear oscillator and a variant of the small twist theorem,” Proc. London Math. Soc., 79, 381–413 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  45. R. Ortega, “Twist mappings, invariant curves and periodic differential equations,” in: Nonlinear Analysis and Its Applications to Differential Equations (Lisbon, 1998), Birkhäuser, Basel (2001), pp. 85–112.

    Google Scholar 

  46. R. Ortega, “Invariant curves of mappings with averaged small twist,” Adv. Nonlin. Stud., 1, 14–39 (2001).

    MATH  MathSciNet  Google Scholar 

  47. R. Ortega, “Periodic perturbations of an isochronous center,” Qual. Theory Dynam. Syst., 3, 83–91 (2002).

    MATH  MathSciNet  Google Scholar 

  48. G. Seifert, “Resonance in undamped second-order nonlinear equations with periodic forcing,” Quart. Appl. Math., 48, 527–530 (1990).

    MATH  MathSciNet  Google Scholar 

  49. Dingbian Qian, “Resonance phenomena for asymmetric weakly nonlinear oscillator,” Sci. China Ser. A, 45, 214–222 (2002).

    MATH  MathSciNet  Google Scholar 

  50. Zaihong Wang, “Multiplicity of periodic solutions of Duffing equations with jumping nonlinearities,” Acta Math. Appl. Sin. Engl. Ser., 18, 513–522 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  51. Zaihong Wang, “Existence and multiplicity of periodic solutions of the second-order differential equations with jumping nonlinearities,” Acta Math. Appl. Sin. Engl. Ser., 615–624.

  52. Zaihong Wang, “Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives,” Discrete Contin. Dynam. Syst., 9, 751–770 (2003).

    MATH  Google Scholar 

  53. Zaihong Wang, “Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities,” Proc. Amer. Math. Soc., 131, 523–531 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  54. Xiaojing Yang, “Unboundedness of the large solutions of some asymmetric oscillators at resonance,” Math. Nachr., 276, 89–102 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  55. Xiaojing Yang, “Unbounded solutions in asymmetric oscillations,” Math. Comput. Modelling, 40, 57–62 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  56. Xiaojing Yang, “Unbounded solutions of asymmetric oscillator,” Math. Proc. Cambridge Phil. Soc., 137, 487–494 (2004).

    Article  MATH  Google Scholar 

  57. Xiaojing Yang, “Existence of periodic solutions in nonlinear asymmetric oscillations,” Bull. London Math. Soc., 37, 566–574 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  58. Xiaojing Yang, “Unbounded solutions of differential equations of second order,” Arch. Math. (Basel), 85, 460–469 (2005).

    MATH  MathSciNet  Google Scholar 

  59. Xiaojing Yang, “Existence of periodic solutions of a class of planar systems,” Z. Anal. Anwend, 25, 237–248 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  60. Xiaojing Yang, Boundedness in Asymmetric Oscillators (to appear).

  61. Xiaomei Zhang and Dingbian Qian, “Existence of periodic solutions for second-order differential equations with asymmetric nonlinearity,” Acta Math. Sinica (Chin. Ser.), 46, 1017–1024 (2003).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    J. Mawhin

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  1. J. Mawhin
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 190–205, February, 2007.

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Mawhin, J. Resonance and nonlinearity: A survey. Ukr Math J 59, 197–214 (2007). https://doi.org/10.1007/s11253-007-0016-1

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  • DOI: https://doi.org/10.1007/s11253-007-0016-1

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